From 2275720c23023433cfcf59b24ffac540aa4dafde Mon Sep 17 00:00:00 2001 From: Jochen <97750753+Jochengehtab@users.noreply.github.com> Date: Fri, 10 Jul 2026 17:10:21 +0200 Subject: [PATCH 1/5] add --- docs/algorithm/add.md | 11 ++ docs/algorithm/div.md | 166 ++++++++++++++++++ .../img/cpp_registermodell_sequentiell.png | Bin 0 -> 410250 bytes ...pp_ueberfuehrung_bytearray_sequentiell.png | Bin 0 -> 455850 bytes docs/algorithm/img/dezimales_modell.png | Bin 0 -> 251161 bytes .../img/teilprodukte_alignment_128bit.png | Bin 0 -> 182703 bytes docs/algorithm/mul.md | 31 ++++ docs/algorithm/sub.md | 7 + docs/bigint.md | 63 +++---- 9 files changed, 247 insertions(+), 31 deletions(-) create mode 100644 docs/algorithm/add.md create mode 100644 docs/algorithm/div.md create mode 100644 docs/algorithm/img/cpp_registermodell_sequentiell.png create mode 100644 docs/algorithm/img/cpp_ueberfuehrung_bytearray_sequentiell.png create mode 100644 docs/algorithm/img/dezimales_modell.png create mode 100644 docs/algorithm/img/teilprodukte_alignment_128bit.png create mode 100644 docs/algorithm/mul.md create mode 100644 docs/algorithm/sub.md diff --git a/docs/algorithm/add.md b/docs/algorithm/add.md new file mode 100644 index 0000000..4089a2a --- /dev/null +++ b/docs/algorithm/add.md @@ -0,0 +1,11 @@ +# Addition +The addition algorithm operates in Base $2^{64}$, using 64-bit unsigned integers (`uint64_t`) to represent each limb of the arbitrary-precision integer. Overflows beyond the 64-bit boundary are detected using standard unsigned arithmetic comparisons. + +1. **Iteration:** The algorithm traverses from the least significant limb (LSB, `index 0`) to the most significant limb (MSB, `index N`). +2. **Carry Propagation:** For each limb, represented by `i`, it calculates: + `sum = aVal + bVal` + An overflow is detected if the intermediate `sum` is smaller than `aVal`. +3. **Carry Handling:** + * The carry for the next iteration (`nextCarry`) is initially flagged as `1` if an overflow occurred in the initial sum. + * The carry from the previous iteration is then added to the `sum`. If this addition causes a secondary overflow (detected by checking `sum < carry`), `nextCarry` is incremented. +4. **Finalization:** If a carry remains after the final limb is processed, an additional limb containing the value `1` is appended to the vector. Finally, the vector is normalized to prune trailing zeros. \ No newline at end of file diff --git a/docs/algorithm/div.md b/docs/algorithm/div.md new file mode 100644 index 0000000..aa66016 --- /dev/null +++ b/docs/algorithm/div.md @@ -0,0 +1,166 @@ +# Division +This documentation explains how the **Bitwise Long Division** (Binary Long Division) is implemented. + +## Definitions +* **Dividend (`data`):** The number being divided. +* **Divisor (`divisor`):** The number you are dividing by. +* **Quotient (`quotient`):** This is the result number. How many times the divisor fits entirely into the dividend. +* **Remainder / Modulo (`remaining`):** The leftover amount that is strictly less than the divisor. +* **Dividend Mask (`dividendMask`):** The active "working remainder" used dynamically within the long division loop to evaluate the current subset of bits. + +## The Basic Concept +Binary Long Division works exactly like the long division taught in elementary school, but it is simpler because the quotient at any step can only ever be `1` or `0` (it either fits, or it doesn't). +1. Isolate the highest piece of the dividend. +2. Check if the divisor fits into it. +3. If it fits, write `1` to the quotient, subtract the divisor, and "bring down" the next bit. +4. If it doesn't fit, write `0` to the quotient, don't subtract, and "bring down" the next bit. + +**Quick Binary Example `7 / 2`:** +* `7` (`111`) +* `2` (`10`) + +**The Setup:** +```text + [Quotient] + _____________ +Divisor: 10 | 1 1 1 <-- Dividend + ^ ^ + MSB LSB + (Highest) (Lowest) +``` + +**Step 1: Isolate the highest bit (MSB)** +```text + 0 <-- 10 does not fit into 1. Write 0 to quotient. + _______ + 10 | 1 1 1 + - 0 <-- No subtraction occurs. + --- + 1 <-- Working remainder is 1. +``` + +**Step 2: Bring down the next bit** +```text + 0 1 <-- 10 fits into 11! Write 1 to quotient. + _______ + 10 | 1 1 1 + . | + 1 1 <-- Bring down the 1. Working remainder is now 11. + - 1 0 <-- Subtract divisor (10). + ----- + 1 <-- New working remainder is 1. +``` + +**Step 3: Bring down the last bit (LSB)** +```text + 0 1 1 <-- 10 fits into 11! Write 1 to quotient. + _______ + 10 | 1 1 1 + . | + 1 1 <-- Bring down the final 1. Working remainder is 11. + - 1 0 <-- Subtract divisor (10). + ----- + 1 <-- Final Remainder! +``` + +**Result:** +The quotient is `001` (`3` in decimal). The leftover value at the bottom is our remainder: **`1`**. +*(7 / 2 = 3 R 1)*. + + +## Implementation +The division algorithm is implemented within 5 steps. + +### 1. Initialization +1. **Division by Zero:** If the divisor is `0` or the dividend is `0` (yielding an invalid MSB index), we abort and default the quotient and remainder to `0`. +2. **Finding the MSB:** Using the `getStartBitIndex` function, we get the absolute highest active bit index across the 64-bit limb array and assign it to `initialDividendIndex`. This ensures that we skip processing leading zero-limbs or "ghost" zeros. +3. **Preallocation:** Because the exact bit-length of the quotient correlates directly to the `initialDividendIndex`, the `quotient` vector is pre-allocated to the maximum required size: `(initialDividendIndex / 64) + 1` 64-bit limbs. + +### 2. Working Remainder (`dividendMask`) +`dividendMask` is initialized by taking a value of `0` and using the `addBitFromNumber` helper to copy down the absolute highest bit from the **dividend** at `initialDividendIndex`. + +### 3. Bitwise Evaluation Loop +In a loop, bit by bit is processed from `initialDividendIndex` down to `-1`. +For every bit position, the mathematical power index (`currentQBitIndex`) is evaluated. + +At each step, we check if the `dividendMask` (working remainder) is $\ge$ the `divisor`. + +**Case A: The Mask is $\ge$ the Divisor** +* The divisor "fits" inside the working remainder. +* A `1` is written to the exact corresponding bit inside the pre-allocated `quotient` vector using bitwise operations: `quotient[currentQBitIndex / 64] |= (1ULL << (currentQBitIndex % 64))`. +* If we have reached the end of the dividend (`dividendIndex < 0`), the final modulo subtraction occurs, and the loop terminates. +* Otherwise, the `divisor` is subtracted from the `dividendMask` in-place, the next bit from the dividend is appended to the mask using `addBitFromNumberInPlace`, and `dividendIndex` is decremented. + +**Case B: The Mask is $<$ the Divisor** +* The divisor does not fit. The quotient bit remains `0` (its default pre-allocated state). +* No subtraction occurs. +* If we have reached the end of the dividend (`dividendIndex < 0`), the `dividendMask` is preserved as-is as the remainder, and the loop terminates. +* Otherwise, the next bit from the dividend is appended to the mask using `addBitFromNumberInPlace` to increase its value for the next loop iteration, and `dividendIndex` is decremented. + +### 4. Modulo (The Remainder) +Because this is integer division, there is often a fractional remainder. The `div` function accepts an optional pointer to a `remaining` vector (`ByteArray* remaining`). +When the loop finishes processing the final bit (`dividendIndex < 0`), whatever mathematical value is left inside the `dividendMask` represents the modulo. If the pointer is provided, the mask (or subtracted mask) is copied into it. + +*(Note: Because of this architecture, evaluating `A % B` requires the same computational effort as `A / B`. Therefore, if both the quotient and remainder are needed, they are extracted simultaneously).* + +### 5. Final Normalization +Even though pre-allocation is tightly bound to the `initialDividendIndex`, the final quotient might have leading zeros depending on the magnitude of the divisor. The `div` function concludes by stripping any trailing zero-limbs from the little-endian vector using `normalizeVector` to maintain strict Base $2^{64}$ normalization guarantees. + +--- + +## Visualization: `13 / 3` +* **Dividend:** `13` (`1101`) +* **Divisor:** `3` (`0011`) + +**Initialization:** +* `getStartBitIndex(1101)` returns `3` (the 0-based index of the highest `1` bit). +* `dividendMask` (the working remainder) is initialized to `0000`. +* `quotient` is pre-allocated and initialized to `0000`. + +### Step 1: Processing Bit Index 3 +* **Action:** Shift `dividendMask` left by 1, and bring down Bit 3 of the dividend (`1`). +* **Mask Status:** `dividendMask` becomes `0001` (Decimal: 1). +* **Comparison:** Is `0001` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** +* **Result:** + * Divisor does not fit. No subtraction. + * Bit 3 of `quotient` remains `0`. + * **Current Quotient:** `0000` + * **Current Mask:** `0001` + +### Step 2: Processing Bit Index 2 +* **Action:** Shift `dividendMask` left by 1 (`0001` $\rightarrow$ `0010`), and bring down Bit 2 of the dividend (`1`). +* **Mask Status:** `dividendMask` becomes `0011` (Decimal: 3). +* **Comparison:** Is `0011` $\ge$ `0011` (Divisor)? $\rightarrow$ **TRUE** +* **Result:** + * Divisor fits! + * Set Bit 2 of `quotient` to `1` using bitwise OR. + * Subtract divisor from mask: `0011` - `0011` = `0000`. + * **Current Quotient:** `0100` + * **Current Mask:** `0000` + +### Step 3: Processing Bit Index 1 +* **Action:** Shift `dividendMask` left by 1 (`0000` $\rightarrow$ `0000`), and bring down Bit 1 of the dividend (`0`). +* **Mask Status:** `dividendMask` becomes `0000` (Decimal: 0). +* **Comparison:** Is `0000` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** +* **Result:** + * Divisor does not fit. No subtraction. + * Bit 1 of `quotient` remains `0`. + * **Current Quotient:** `0100` + * **Current Mask:** `0000` + +### Step 4: Processing Bit Index 0 (LSB) +* **Action:** Shift `dividendMask` left by 1 (`0000` $\rightarrow$ `0000`), and bring down Bit 0 of the dividend (`1`). +* **Mask Status:** `dividendMask` becomes `0001` (Decimal: 1). +* **Comparison:** Is `0001` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** +* **Result:** + * Divisor does not fit. No subtraction. + * Bit 0 of `quotient` remains `0`. + * **Current Quotient:** `0100` + * **Current Mask:** `0001` + +### Final Output Evaluation: +The loop terminates because we have processed bit index `0`. +1. **The Quotient:** The pre-allocated quotient vector holds `0100`, which mathematically evaluates to **`4`**. +2. **The Remainder:** The `dividendMask` is left holding `0001`, which mathematically evaluates to **`1`**. 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It is designed to handle infinitely large unsigned integers by utilizing dynamically expanding contiguous memory (`std::vector`). -By operating entirely in software, `BigInt` bypasses hardware ALU constraints (like standard 64-bit boundaries), making it suitable for cryptographic computations where numbers routinely span thousands of bits. +By operating entirely in software, `Base256` bypasses hardware ALU constraints (like standard 64-bit boundaries), making it suitable for cryptographic computations where numbers routinely span thousands of bits. --- ## Technical Architecture -### 1. The 64-Bit Limb System -Hardware natively computes in Base-2 (Binary), while human-readable formats use Base-10 (Decimal). This class uses a base-2^64 representation, treating exactly one `uint64_t` limb as a single, discrete digit. +### 1. The Radix-256 (Base-256) System +Hardware natively computes in Base-2 (Binary), while human-readable formats use Base-10 (Decimal). This class uses **Base-256**, treating exactly **one byte** (`uint8_t`) as a single, discrete "digit." -Each digit holds a value from `0` to `2^64 - 1`. If an arithmetic operation pushes a limb beyond that range, it overflows and carries into the next limb. The total mathematical value is represented by the polynomial: +Each digit holds a value from `0` to `255`. If an arithmetic operation pushes a byte beyond 255, it overflows and carries a `1` into the next byte magnitude. The total mathematical value is represented by the polynomial: -`Value = d[0]*((2^64)^0) + d[1]*((2^64)^1) + d[2]*((2^64)^2) ... + d[n]*((2^64)^n)` +`Value = d[0]*(256^0) + d[1]*(256^1) + d[2]*(256^2) ... + d[n]*(256^n)` ### 2. Little-Endian Memory Mapping -Numbers are stored in **little-endian** order, meaning the least-significant limb is stored at index `0` of the vector. -Or in other words `(2^64)^0` is always stored at `data[0]`. +Numbers are stored in **little-endian** order, meaning the least-significant byte (LSB) is stored at index `0` of the vector. +Or in other words 256^0 is always stored at `data[0]`. -This architecture guarantees that the array index maps perfectly to the radix exponent. The limb at `data[i]` is strictly bound to `(2^64)^i`. This eliminates the need for complex index inversion during iterative algorithms. +This architecture guarantees that the array index maps perfectly to the radix exponent. The byte at `data[i]` is strictly bound to `256^i`. This eliminates the need for complex index inversion during iterative algorithms. -| Number (Base 10) | Hexadecimal | Limb Array `vector` | Radix Evaluation | -|:-----------------|:-----------------------|:------------------------------|:-----------------------------------------| -| `42` | `0x2A` | `[42]` | 42 * (2^64)^0 | -| `2^64` | `0x010000000000000000` | `[0, 1]` | 0 * (2^64)^0 + 1 * (2^64)^1 | -| `2^64 + 7` | `0x010000000000000007` | `[7, 1]` | 7 * (2^64)^0 + 1 * (2^64)^1 | +| Number (Base 10) | Hexadecimal | Base-256 Array `vector` | Radix Evaluation | +|:-----------------|:------------|:---------------------------------|:----------------------------------| +| `42` | `0x2A` | `[42]` | 42 * 256^0 | +| `258` | `0x0102` | `[2, 1]` | 2 * 256^0 + 1 * 256^1 | +| `65536` | `0x010000` | `[0, 0, 1]` | 0 * 256^0 + 0 * 256^1 + 1 * 256^2 | ### 3. Vector Normalization Because operations like subtraction and division can shrink the magnitude of a number, mathematical leading zeros appear as trailing elements at the end of the little-endian vector. -The class enforces strict **Array Normalization**. After every operation, trailing zero-limbs are instantly popped from the vector until only the true magnitude remains (or until a single `[0]` is left). This ensures `data.size()` operates as a reliable magnitude heuristic and prevents logic failure during bitwise comparisons. +The class enforces strict **Array Normalization**. After every operation, trailing zero-bytes are instantly popped from the vector until only the true magnitude remains (or until a single `[0]` is left). This ensures `data.size()` operates as a reliable magnitude heuristic and prevents logic failure during bitwise comparisons. --- ## Algorithmic Mechanics -The implementation provides addition, subtraction, multiplication, division, modulo, comparisons, decimal output, and selected helper functions in `operations::math`. - +- [Addition](algorithms/add.md#addition) +- [Subtraction](algorithms/sub.md#subtraction) +- [Subtraction](algorithms/mul.md#multiplication) +- [Subtraction](algorithms/div.md#division) --- ## Code Examples & Usage -The class acts like an unsigned primitive integer, natively supporting cross-limb cascading, memory-safe aliasing, and underflow clamping. +The class acts identically to standard primitive integers, natively supporting cross-byte cascading, memory-safe aliasing, and underflow clamping. ```cpp -#include "bigint.h" - -using operations::BigInt; +#include "vec/operations.h" +using operations::Base256; // 1. Initialization -BigInt a(4294967295); // Initializes from max 32-bit int -BigInt b(2); +Base256 a(4294967295); // Initializes from max 32-bit int +Base256 b(2); // 2. Infinite Precision Arithmetic -BigInt sum = a + b; // Handles limb-overflows natively -BigInt prod = a * b; // Expands underlying vector memory dynamically -BigInt quot = a / b; // Evaluates using bitwise long-division +Base256 sum = a + b; // Handles byte-overflows natively +Base256 prod = a * b; // Expands underlying vector memory dynamically +Base256 quot = a / b; // Evaluates using bitwise long-division // 3. Safe Compound Assignment & Aliasing a += b; a *= a; // Memory-safe aliasing (reads/writes safely isolated) // 4. Edge-Case Safety -BigInt zero = b - a; // Negative subtraction cleanly clamps to 0 -BigInt drop = b / a; // Fractional division drops remainder (evaluates to 0) +Base256 zero = b - a; // Negative subtraction cleanly clamps to 0 +Base256 drop = b / a; // Fractional division drops remainder (evaluates to 0) // 5. Comparisons & Output if (sum > a && prod != zero) { - sum.print(); // Iterates in reverse to print standard big-endian output + sum.print(); // Iterates in reverse to print standard Big-Endian output } -``` +``` \ No newline at end of file From b1ed80c004c786d120be3c1c9c6d579b2dba4f5a Mon Sep 17 00:00:00 2001 From: Jochen <97750753+Jochengehtab@users.noreply.github.com> Date: Fri, 10 Jul 2026 17:48:56 +0200 Subject: [PATCH 2/5] Revert "add" This reverts commit 2275720c23023433cfcf59b24ffac540aa4dafde. --- docs/algorithm/add.md | 11 -- docs/algorithm/div.md | 166 ------------------ .../img/cpp_registermodell_sequentiell.png | Bin 410250 -> 0 bytes ...pp_ueberfuehrung_bytearray_sequentiell.png | Bin 455850 -> 0 bytes docs/algorithm/img/dezimales_modell.png | Bin 251161 -> 0 bytes .../img/teilprodukte_alignment_128bit.png | Bin 182703 -> 0 bytes docs/algorithm/mul.md | 31 ---- docs/algorithm/sub.md | 7 - docs/bigint.md | 63 ++++--- 9 files changed, 31 insertions(+), 247 deletions(-) delete mode 100644 docs/algorithm/add.md delete mode 100644 docs/algorithm/div.md delete mode 100644 docs/algorithm/img/cpp_registermodell_sequentiell.png delete mode 100644 docs/algorithm/img/cpp_ueberfuehrung_bytearray_sequentiell.png delete mode 100644 docs/algorithm/img/dezimales_modell.png delete mode 100644 docs/algorithm/img/teilprodukte_alignment_128bit.png delete mode 100644 docs/algorithm/mul.md delete mode 100644 docs/algorithm/sub.md diff --git a/docs/algorithm/add.md b/docs/algorithm/add.md deleted file mode 100644 index 4089a2a..0000000 --- a/docs/algorithm/add.md +++ /dev/null @@ -1,11 +0,0 @@ -# Addition -The addition algorithm operates in Base $2^{64}$, using 64-bit unsigned integers (`uint64_t`) to represent each limb of the arbitrary-precision integer. Overflows beyond the 64-bit boundary are detected using standard unsigned arithmetic comparisons. - -1. **Iteration:** The algorithm traverses from the least significant limb (LSB, `index 0`) to the most significant limb (MSB, `index N`). -2. **Carry Propagation:** For each limb, represented by `i`, it calculates: - `sum = aVal + bVal` - An overflow is detected if the intermediate `sum` is smaller than `aVal`. -3. **Carry Handling:** - * The carry for the next iteration (`nextCarry`) is initially flagged as `1` if an overflow occurred in the initial sum. - * The carry from the previous iteration is then added to the `sum`. If this addition causes a secondary overflow (detected by checking `sum < carry`), `nextCarry` is incremented. -4. **Finalization:** If a carry remains after the final limb is processed, an additional limb containing the value `1` is appended to the vector. Finally, the vector is normalized to prune trailing zeros. \ No newline at end of file diff --git a/docs/algorithm/div.md b/docs/algorithm/div.md deleted file mode 100644 index aa66016..0000000 --- a/docs/algorithm/div.md +++ /dev/null @@ -1,166 +0,0 @@ -# Division -This documentation explains how the **Bitwise Long Division** (Binary Long Division) is implemented. - -## Definitions -* **Dividend (`data`):** The number being divided. -* **Divisor (`divisor`):** The number you are dividing by. -* **Quotient (`quotient`):** This is the result number. How many times the divisor fits entirely into the dividend. -* **Remainder / Modulo (`remaining`):** The leftover amount that is strictly less than the divisor. -* **Dividend Mask (`dividendMask`):** The active "working remainder" used dynamically within the long division loop to evaluate the current subset of bits. - -## The Basic Concept -Binary Long Division works exactly like the long division taught in elementary school, but it is simpler because the quotient at any step can only ever be `1` or `0` (it either fits, or it doesn't). -1. Isolate the highest piece of the dividend. -2. Check if the divisor fits into it. -3. If it fits, write `1` to the quotient, subtract the divisor, and "bring down" the next bit. -4. If it doesn't fit, write `0` to the quotient, don't subtract, and "bring down" the next bit. - -**Quick Binary Example `7 / 2`:** -* `7` (`111`) -* `2` (`10`) - -**The Setup:** -```text - [Quotient] - _____________ -Divisor: 10 | 1 1 1 <-- Dividend - ^ ^ - MSB LSB - (Highest) (Lowest) -``` - -**Step 1: Isolate the highest bit (MSB)** -```text - 0 <-- 10 does not fit into 1. Write 0 to quotient. - _______ - 10 | 1 1 1 - - 0 <-- No subtraction occurs. - --- - 1 <-- Working remainder is 1. -``` - -**Step 2: Bring down the next bit** -```text - 0 1 <-- 10 fits into 11! Write 1 to quotient. - _______ - 10 | 1 1 1 - . | - 1 1 <-- Bring down the 1. Working remainder is now 11. - - 1 0 <-- Subtract divisor (10). - ----- - 1 <-- New working remainder is 1. -``` - -**Step 3: Bring down the last bit (LSB)** -```text - 0 1 1 <-- 10 fits into 11! Write 1 to quotient. - _______ - 10 | 1 1 1 - . | - 1 1 <-- Bring down the final 1. Working remainder is 11. - - 1 0 <-- Subtract divisor (10). - ----- - 1 <-- Final Remainder! -``` - -**Result:** -The quotient is `001` (`3` in decimal). The leftover value at the bottom is our remainder: **`1`**. -*(7 / 2 = 3 R 1)*. - - -## Implementation -The division algorithm is implemented within 5 steps. - -### 1. Initialization -1. **Division by Zero:** If the divisor is `0` or the dividend is `0` (yielding an invalid MSB index), we abort and default the quotient and remainder to `0`. -2. **Finding the MSB:** Using the `getStartBitIndex` function, we get the absolute highest active bit index across the 64-bit limb array and assign it to `initialDividendIndex`. This ensures that we skip processing leading zero-limbs or "ghost" zeros. -3. **Preallocation:** Because the exact bit-length of the quotient correlates directly to the `initialDividendIndex`, the `quotient` vector is pre-allocated to the maximum required size: `(initialDividendIndex / 64) + 1` 64-bit limbs. - -### 2. Working Remainder (`dividendMask`) -`dividendMask` is initialized by taking a value of `0` and using the `addBitFromNumber` helper to copy down the absolute highest bit from the **dividend** at `initialDividendIndex`. - -### 3. Bitwise Evaluation Loop -In a loop, bit by bit is processed from `initialDividendIndex` down to `-1`. -For every bit position, the mathematical power index (`currentQBitIndex`) is evaluated. - -At each step, we check if the `dividendMask` (working remainder) is $\ge$ the `divisor`. - -**Case A: The Mask is $\ge$ the Divisor** -* The divisor "fits" inside the working remainder. -* A `1` is written to the exact corresponding bit inside the pre-allocated `quotient` vector using bitwise operations: `quotient[currentQBitIndex / 64] |= (1ULL << (currentQBitIndex % 64))`. -* If we have reached the end of the dividend (`dividendIndex < 0`), the final modulo subtraction occurs, and the loop terminates. -* Otherwise, the `divisor` is subtracted from the `dividendMask` in-place, the next bit from the dividend is appended to the mask using `addBitFromNumberInPlace`, and `dividendIndex` is decremented. - -**Case B: The Mask is $<$ the Divisor** -* The divisor does not fit. The quotient bit remains `0` (its default pre-allocated state). -* No subtraction occurs. -* If we have reached the end of the dividend (`dividendIndex < 0`), the `dividendMask` is preserved as-is as the remainder, and the loop terminates. -* Otherwise, the next bit from the dividend is appended to the mask using `addBitFromNumberInPlace` to increase its value for the next loop iteration, and `dividendIndex` is decremented. - -### 4. Modulo (The Remainder) -Because this is integer division, there is often a fractional remainder. The `div` function accepts an optional pointer to a `remaining` vector (`ByteArray* remaining`). -When the loop finishes processing the final bit (`dividendIndex < 0`), whatever mathematical value is left inside the `dividendMask` represents the modulo. If the pointer is provided, the mask (or subtracted mask) is copied into it. - -*(Note: Because of this architecture, evaluating `A % B` requires the same computational effort as `A / B`. Therefore, if both the quotient and remainder are needed, they are extracted simultaneously).* - -### 5. Final Normalization -Even though pre-allocation is tightly bound to the `initialDividendIndex`, the final quotient might have leading zeros depending on the magnitude of the divisor. The `div` function concludes by stripping any trailing zero-limbs from the little-endian vector using `normalizeVector` to maintain strict Base $2^{64}$ normalization guarantees. - ---- - -## Visualization: `13 / 3` -* **Dividend:** `13` (`1101`) -* **Divisor:** `3` (`0011`) - -**Initialization:** -* `getStartBitIndex(1101)` returns `3` (the 0-based index of the highest `1` bit). -* `dividendMask` (the working remainder) is initialized to `0000`. -* `quotient` is pre-allocated and initialized to `0000`. - -### Step 1: Processing Bit Index 3 -* **Action:** Shift `dividendMask` left by 1, and bring down Bit 3 of the dividend (`1`). -* **Mask Status:** `dividendMask` becomes `0001` (Decimal: 1). -* **Comparison:** Is `0001` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** -* **Result:** - * Divisor does not fit. No subtraction. - * Bit 3 of `quotient` remains `0`. - * **Current Quotient:** `0000` - * **Current Mask:** `0001` - -### Step 2: Processing Bit Index 2 -* **Action:** Shift `dividendMask` left by 1 (`0001` $\rightarrow$ `0010`), and bring down Bit 2 of the dividend (`1`). -* **Mask Status:** `dividendMask` becomes `0011` (Decimal: 3). -* **Comparison:** Is `0011` $\ge$ `0011` (Divisor)? $\rightarrow$ **TRUE** -* **Result:** - * Divisor fits! - * Set Bit 2 of `quotient` to `1` using bitwise OR. - * Subtract divisor from mask: `0011` - `0011` = `0000`. - * **Current Quotient:** `0100` - * **Current Mask:** `0000` - -### Step 3: Processing Bit Index 1 -* **Action:** Shift `dividendMask` left by 1 (`0000` $\rightarrow$ `0000`), and bring down Bit 1 of the dividend (`0`). -* **Mask Status:** `dividendMask` becomes `0000` (Decimal: 0). -* **Comparison:** Is `0000` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** -* **Result:** - * Divisor does not fit. No subtraction. - * Bit 1 of `quotient` remains `0`. - * **Current Quotient:** `0100` - * **Current Mask:** `0000` - -### Step 4: Processing Bit Index 0 (LSB) -* **Action:** Shift `dividendMask` left by 1 (`0000` $\rightarrow$ `0000`), and bring down Bit 0 of the dividend (`1`). -* **Mask Status:** `dividendMask` becomes `0001` (Decimal: 1). -* **Comparison:** Is `0001` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** -* **Result:** - * Divisor does not fit. No subtraction. - * Bit 0 of `quotient` remains `0`. - * **Current Quotient:** `0100` - * **Current Mask:** `0001` - -### Final Output Evaluation: -The loop terminates because we have processed bit index `0`. -1. **The Quotient:** The pre-allocated quotient vector holds `0100`, which mathematically evaluates to **`4`**. -2. **The Remainder:** The `dividendMask` is left holding `0001`, which mathematically evaluates to **`1`**. 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zgr$nHrHR@?G-C_g5eJy+8NCKZG)&Q`=KwzraK&DV3l}U;QemZ~SW1kChwe9K3hI%u*T~v#(U% za7af7li&uIM!r{;W%8B`?;cI zY2f(84Mv3;$Md^F=#9KM2H7Qk|HpQ*pX)_VoCtNbRO$Wu`0ZG-vtwqT07e?v?FScs z7`m9)x?71MSwHSDjORcpVK-)hlDoaA*jn} z<(JYOKU&c?8Dz3;TjSpgJF>Ia>o)D&^-uL(Uj!47l6RCF;(u@;xh~vswl*^Q4_EQF zv-7#ywLrl5z}k;}(MERRa5CT3iX8Z{#+j~}p6AUtFGxAD>xzx-XEfj&0M`m1OHeFyuu$?=bMn&%6{ zVC{>bA4P3K@r(@865|;$a-D9z3AP3f#{EzAx4yhFG1AEK+zX_R5dBby0R5sVKoz9l z>nZ=KaP)IkmZ51DS`C&uzmb3ZSjKzV61=oDHg4wgm%PjWK4ZRhz=VF`ztzC`b1kZo z5$}wmbMvy-bl;)SFsOm^=Q28|*!Wv@?dSajS`l3tcDDa_A9OtUJLbQ@_6kGG>4(2F z@cAK6{_P>B#>iCtLoe|E#ytN#$+CgthFk|=&<>#iEjI%LsHFRiDio@q_8Xt8j=}El z16wKN9|4_yxaXK?-&ga7{$NdD`S w9QMQWpSC;qMT7XCwE{-q|Apz`y~5da{gCCg$F3fQ&@<3Jt9K^`). +The **`BigInt`** class is a dynamically sized, arbitrary-precision integer implementation. It is designed to handle infinitely large unsigned integers by utilizing dynamically expanding contiguous memory (`std::vector`). -By operating entirely in software, `Base256` bypasses hardware ALU constraints (like standard 64-bit boundaries), making it suitable for cryptographic computations where numbers routinely span thousands of bits. +By operating entirely in software, `BigInt` bypasses hardware ALU constraints (like standard 64-bit boundaries), making it suitable for cryptographic computations where numbers routinely span thousands of bits. --- ## Technical Architecture -### 1. The Radix-256 (Base-256) System -Hardware natively computes in Base-2 (Binary), while human-readable formats use Base-10 (Decimal). This class uses **Base-256**, treating exactly **one byte** (`uint8_t`) as a single, discrete "digit." +### 1. The 64-Bit Limb System +Hardware natively computes in Base-2 (Binary), while human-readable formats use Base-10 (Decimal). This class uses a base-2^64 representation, treating exactly one `uint64_t` limb as a single, discrete digit. -Each digit holds a value from `0` to `255`. If an arithmetic operation pushes a byte beyond 255, it overflows and carries a `1` into the next byte magnitude. The total mathematical value is represented by the polynomial: +Each digit holds a value from `0` to `2^64 - 1`. If an arithmetic operation pushes a limb beyond that range, it overflows and carries into the next limb. The total mathematical value is represented by the polynomial: -`Value = d[0]*(256^0) + d[1]*(256^1) + d[2]*(256^2) ... + d[n]*(256^n)` +`Value = d[0]*((2^64)^0) + d[1]*((2^64)^1) + d[2]*((2^64)^2) ... + d[n]*((2^64)^n)` ### 2. Little-Endian Memory Mapping -Numbers are stored in **little-endian** order, meaning the least-significant byte (LSB) is stored at index `0` of the vector. -Or in other words 256^0 is always stored at `data[0]`. +Numbers are stored in **little-endian** order, meaning the least-significant limb is stored at index `0` of the vector. +Or in other words `(2^64)^0` is always stored at `data[0]`. -This architecture guarantees that the array index maps perfectly to the radix exponent. The byte at `data[i]` is strictly bound to `256^i`. This eliminates the need for complex index inversion during iterative algorithms. +This architecture guarantees that the array index maps perfectly to the radix exponent. The limb at `data[i]` is strictly bound to `(2^64)^i`. This eliminates the need for complex index inversion during iterative algorithms. -| Number (Base 10) | Hexadecimal | Base-256 Array `vector` | Radix Evaluation | -|:-----------------|:------------|:---------------------------------|:----------------------------------| -| `42` | `0x2A` | `[42]` | 42 * 256^0 | -| `258` | `0x0102` | `[2, 1]` | 2 * 256^0 + 1 * 256^1 | -| `65536` | `0x010000` | `[0, 0, 1]` | 0 * 256^0 + 0 * 256^1 + 1 * 256^2 | +| Number (Base 10) | Hexadecimal | Limb Array `vector` | Radix Evaluation | +|:-----------------|:-----------------------|:------------------------------|:-----------------------------------------| +| `42` | `0x2A` | `[42]` | 42 * (2^64)^0 | +| `2^64` | `0x010000000000000000` | `[0, 1]` | 0 * (2^64)^0 + 1 * (2^64)^1 | +| `2^64 + 7` | `0x010000000000000007` | `[7, 1]` | 7 * (2^64)^0 + 1 * (2^64)^1 | ### 3. Vector Normalization Because operations like subtraction and division can shrink the magnitude of a number, mathematical leading zeros appear as trailing elements at the end of the little-endian vector. -The class enforces strict **Array Normalization**. After every operation, trailing zero-bytes are instantly popped from the vector until only the true magnitude remains (or until a single `[0]` is left). This ensures `data.size()` operates as a reliable magnitude heuristic and prevents logic failure during bitwise comparisons. +The class enforces strict **Array Normalization**. After every operation, trailing zero-limbs are instantly popped from the vector until only the true magnitude remains (or until a single `[0]` is left). This ensures `data.size()` operates as a reliable magnitude heuristic and prevents logic failure during bitwise comparisons. --- ## Algorithmic Mechanics -- [Addition](algorithms/add.md#addition) -- [Subtraction](algorithms/sub.md#subtraction) -- [Subtraction](algorithms/mul.md#multiplication) -- [Subtraction](algorithms/div.md#division) +The implementation provides addition, subtraction, multiplication, division, modulo, comparisons, decimal output, and selected helper functions in `operations::math`. + --- ## Code Examples & Usage -The class acts identically to standard primitive integers, natively supporting cross-byte cascading, memory-safe aliasing, and underflow clamping. +The class acts like an unsigned primitive integer, natively supporting cross-limb cascading, memory-safe aliasing, and underflow clamping. ```cpp -#include "vec/operations.h" -using operations::Base256; +#include "bigint.h" + +using operations::BigInt; // 1. Initialization -Base256 a(4294967295); // Initializes from max 32-bit int -Base256 b(2); +BigInt a(4294967295); // Initializes from max 32-bit int +BigInt b(2); // 2. Infinite Precision Arithmetic -Base256 sum = a + b; // Handles byte-overflows natively -Base256 prod = a * b; // Expands underlying vector memory dynamically -Base256 quot = a / b; // Evaluates using bitwise long-division +BigInt sum = a + b; // Handles limb-overflows natively +BigInt prod = a * b; // Expands underlying vector memory dynamically +BigInt quot = a / b; // Evaluates using bitwise long-division // 3. Safe Compound Assignment & Aliasing a += b; a *= a; // Memory-safe aliasing (reads/writes safely isolated) // 4. Edge-Case Safety -Base256 zero = b - a; // Negative subtraction cleanly clamps to 0 -Base256 drop = b / a; // Fractional division drops remainder (evaluates to 0) +BigInt zero = b - a; // Negative subtraction cleanly clamps to 0 +BigInt drop = b / a; // Fractional division drops remainder (evaluates to 0) // 5. Comparisons & Output if (sum > a && prod != zero) { - sum.print(); // Iterates in reverse to print standard Big-Endian output + sum.print(); // Iterates in reverse to print standard big-endian output } -``` \ No newline at end of file +``` From de1001abd8bdb5cbb97eb4519e8724cc0dc5aa41 Mon Sep 17 00:00:00 2001 From: Jochen <97750753+Jochengehtab@users.noreply.github.com> Date: Fri, 10 Jul 2026 17:49:27 +0200 Subject: [PATCH 3/5] add --- docs/algorithm/add.md | 11 ++ docs/algorithm/div.md | 166 ++++++++++++++++++ .../img/cpp_registermodell_sequentiell.png | Bin 0 -> 410250 bytes ...pp_ueberfuehrung_bytearray_sequentiell.png | Bin 0 -> 455850 bytes docs/algorithm/img/dezimales_modell.png | Bin 0 -> 251161 bytes .../img/teilprodukte_alignment_128bit.png | Bin 0 -> 182703 bytes docs/algorithm/mul.md | 31 ++++ docs/algorithm/sub.md | 7 + 8 files changed, 215 insertions(+) create mode 100644 docs/algorithm/add.md create mode 100644 docs/algorithm/div.md create mode 100644 docs/algorithm/img/cpp_registermodell_sequentiell.png create mode 100644 docs/algorithm/img/cpp_ueberfuehrung_bytearray_sequentiell.png create mode 100644 docs/algorithm/img/dezimales_modell.png create mode 100644 docs/algorithm/img/teilprodukte_alignment_128bit.png create mode 100644 docs/algorithm/mul.md create mode 100644 docs/algorithm/sub.md diff --git a/docs/algorithm/add.md b/docs/algorithm/add.md new file mode 100644 index 0000000..4089a2a --- /dev/null +++ b/docs/algorithm/add.md @@ -0,0 +1,11 @@ +# Addition +The addition algorithm operates in Base $2^{64}$, using 64-bit unsigned integers (`uint64_t`) to represent each limb of the arbitrary-precision integer. Overflows beyond the 64-bit boundary are detected using standard unsigned arithmetic comparisons. + +1. **Iteration:** The algorithm traverses from the least significant limb (LSB, `index 0`) to the most significant limb (MSB, `index N`). +2. **Carry Propagation:** For each limb, represented by `i`, it calculates: + `sum = aVal + bVal` + An overflow is detected if the intermediate `sum` is smaller than `aVal`. +3. **Carry Handling:** + * The carry for the next iteration (`nextCarry`) is initially flagged as `1` if an overflow occurred in the initial sum. + * The carry from the previous iteration is then added to the `sum`. If this addition causes a secondary overflow (detected by checking `sum < carry`), `nextCarry` is incremented. +4. **Finalization:** If a carry remains after the final limb is processed, an additional limb containing the value `1` is appended to the vector. Finally, the vector is normalized to prune trailing zeros. \ No newline at end of file diff --git a/docs/algorithm/div.md b/docs/algorithm/div.md new file mode 100644 index 0000000..aa66016 --- /dev/null +++ b/docs/algorithm/div.md @@ -0,0 +1,166 @@ +# Division +This documentation explains how the **Bitwise Long Division** (Binary Long Division) is implemented. + +## Definitions +* **Dividend (`data`):** The number being divided. +* **Divisor (`divisor`):** The number you are dividing by. +* **Quotient (`quotient`):** This is the result number. How many times the divisor fits entirely into the dividend. +* **Remainder / Modulo (`remaining`):** The leftover amount that is strictly less than the divisor. +* **Dividend Mask (`dividendMask`):** The active "working remainder" used dynamically within the long division loop to evaluate the current subset of bits. + +## The Basic Concept +Binary Long Division works exactly like the long division taught in elementary school, but it is simpler because the quotient at any step can only ever be `1` or `0` (it either fits, or it doesn't). +1. Isolate the highest piece of the dividend. +2. Check if the divisor fits into it. +3. If it fits, write `1` to the quotient, subtract the divisor, and "bring down" the next bit. +4. If it doesn't fit, write `0` to the quotient, don't subtract, and "bring down" the next bit. + +**Quick Binary Example `7 / 2`:** +* `7` (`111`) +* `2` (`10`) + +**The Setup:** +```text + [Quotient] + _____________ +Divisor: 10 | 1 1 1 <-- Dividend + ^ ^ + MSB LSB + (Highest) (Lowest) +``` + +**Step 1: Isolate the highest bit (MSB)** +```text + 0 <-- 10 does not fit into 1. Write 0 to quotient. + _______ + 10 | 1 1 1 + - 0 <-- No subtraction occurs. + --- + 1 <-- Working remainder is 1. +``` + +**Step 2: Bring down the next bit** +```text + 0 1 <-- 10 fits into 11! Write 1 to quotient. + _______ + 10 | 1 1 1 + . | + 1 1 <-- Bring down the 1. Working remainder is now 11. + - 1 0 <-- Subtract divisor (10). + ----- + 1 <-- New working remainder is 1. +``` + +**Step 3: Bring down the last bit (LSB)** +```text + 0 1 1 <-- 10 fits into 11! Write 1 to quotient. + _______ + 10 | 1 1 1 + . | + 1 1 <-- Bring down the final 1. Working remainder is 11. + - 1 0 <-- Subtract divisor (10). + ----- + 1 <-- Final Remainder! +``` + +**Result:** +The quotient is `001` (`3` in decimal). The leftover value at the bottom is our remainder: **`1`**. +*(7 / 2 = 3 R 1)*. + + +## Implementation +The division algorithm is implemented within 5 steps. + +### 1. Initialization +1. **Division by Zero:** If the divisor is `0` or the dividend is `0` (yielding an invalid MSB index), we abort and default the quotient and remainder to `0`. +2. **Finding the MSB:** Using the `getStartBitIndex` function, we get the absolute highest active bit index across the 64-bit limb array and assign it to `initialDividendIndex`. This ensures that we skip processing leading zero-limbs or "ghost" zeros. +3. **Preallocation:** Because the exact bit-length of the quotient correlates directly to the `initialDividendIndex`, the `quotient` vector is pre-allocated to the maximum required size: `(initialDividendIndex / 64) + 1` 64-bit limbs. + +### 2. Working Remainder (`dividendMask`) +`dividendMask` is initialized by taking a value of `0` and using the `addBitFromNumber` helper to copy down the absolute highest bit from the **dividend** at `initialDividendIndex`. + +### 3. Bitwise Evaluation Loop +In a loop, bit by bit is processed from `initialDividendIndex` down to `-1`. +For every bit position, the mathematical power index (`currentQBitIndex`) is evaluated. + +At each step, we check if the `dividendMask` (working remainder) is $\ge$ the `divisor`. + +**Case A: The Mask is $\ge$ the Divisor** +* The divisor "fits" inside the working remainder. +* A `1` is written to the exact corresponding bit inside the pre-allocated `quotient` vector using bitwise operations: `quotient[currentQBitIndex / 64] |= (1ULL << (currentQBitIndex % 64))`. +* If we have reached the end of the dividend (`dividendIndex < 0`), the final modulo subtraction occurs, and the loop terminates. +* Otherwise, the `divisor` is subtracted from the `dividendMask` in-place, the next bit from the dividend is appended to the mask using `addBitFromNumberInPlace`, and `dividendIndex` is decremented. + +**Case B: The Mask is $<$ the Divisor** +* The divisor does not fit. The quotient bit remains `0` (its default pre-allocated state). +* No subtraction occurs. +* If we have reached the end of the dividend (`dividendIndex < 0`), the `dividendMask` is preserved as-is as the remainder, and the loop terminates. +* Otherwise, the next bit from the dividend is appended to the mask using `addBitFromNumberInPlace` to increase its value for the next loop iteration, and `dividendIndex` is decremented. + +### 4. Modulo (The Remainder) +Because this is integer division, there is often a fractional remainder. The `div` function accepts an optional pointer to a `remaining` vector (`ByteArray* remaining`). +When the loop finishes processing the final bit (`dividendIndex < 0`), whatever mathematical value is left inside the `dividendMask` represents the modulo. If the pointer is provided, the mask (or subtracted mask) is copied into it. + +*(Note: Because of this architecture, evaluating `A % B` requires the same computational effort as `A / B`. Therefore, if both the quotient and remainder are needed, they are extracted simultaneously).* + +### 5. Final Normalization +Even though pre-allocation is tightly bound to the `initialDividendIndex`, the final quotient might have leading zeros depending on the magnitude of the divisor. The `div` function concludes by stripping any trailing zero-limbs from the little-endian vector using `normalizeVector` to maintain strict Base $2^{64}$ normalization guarantees. + +--- + +## Visualization: `13 / 3` +* **Dividend:** `13` (`1101`) +* **Divisor:** `3` (`0011`) + +**Initialization:** +* `getStartBitIndex(1101)` returns `3` (the 0-based index of the highest `1` bit). +* `dividendMask` (the working remainder) is initialized to `0000`. +* `quotient` is pre-allocated and initialized to `0000`. + +### Step 1: Processing Bit Index 3 +* **Action:** Shift `dividendMask` left by 1, and bring down Bit 3 of the dividend (`1`). +* **Mask Status:** `dividendMask` becomes `0001` (Decimal: 1). +* **Comparison:** Is `0001` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** +* **Result:** + * Divisor does not fit. No subtraction. + * Bit 3 of `quotient` remains `0`. + * **Current Quotient:** `0000` + * **Current Mask:** `0001` + +### Step 2: Processing Bit Index 2 +* **Action:** Shift `dividendMask` left by 1 (`0001` $\rightarrow$ `0010`), and bring down Bit 2 of the dividend (`1`). +* **Mask Status:** `dividendMask` becomes `0011` (Decimal: 3). +* **Comparison:** Is `0011` $\ge$ `0011` (Divisor)? $\rightarrow$ **TRUE** +* **Result:** + * Divisor fits! + * Set Bit 2 of `quotient` to `1` using bitwise OR. + * Subtract divisor from mask: `0011` - `0011` = `0000`. + * **Current Quotient:** `0100` + * **Current Mask:** `0000` + +### Step 3: Processing Bit Index 1 +* **Action:** Shift `dividendMask` left by 1 (`0000` $\rightarrow$ `0000`), and bring down Bit 1 of the dividend (`0`). +* **Mask Status:** `dividendMask` becomes `0000` (Decimal: 0). +* **Comparison:** Is `0000` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** +* **Result:** + * Divisor does not fit. No subtraction. + * Bit 1 of `quotient` remains `0`. + * **Current Quotient:** `0100` + * **Current Mask:** `0000` + +### Step 4: Processing Bit Index 0 (LSB) +* **Action:** Shift `dividendMask` left by 1 (`0000` $\rightarrow$ `0000`), and bring down Bit 0 of the dividend (`1`). +* **Mask Status:** `dividendMask` becomes `0001` (Decimal: 1). +* **Comparison:** Is `0001` $\ge$ `0011` (Divisor)? $\rightarrow$ **FALSE** +* **Result:** + * Divisor does not fit. No subtraction. + * Bit 0 of `quotient` remains `0`. + * **Current Quotient:** `0100` + * **Current Mask:** `0001` + +### Final Output Evaluation: +The loop terminates because we have processed bit index `0`. +1. **The Quotient:** The pre-allocated quotient vector holds `0100`, which mathematically evaluates to **`4`**. +2. **The Remainder:** The `dividendMask` is left holding `0001`, which mathematically evaluates to **`1`**. 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-1,6 +1,6 @@ # Multiplication Algorithm -This document outlines the sequential multiplication logic implemented in the `Base256` class. +This document outlines the sequential multiplication logic implemented in the `BigInt` class. --- From 1585e41966ed10193aa58bd9d9d0f5fa2729ea3b Mon Sep 17 00:00:00 2001 From: Jochen <97750753+Jochengehtab@users.noreply.github.com> Date: Sat, 11 Jul 2026 11:12:36 +0200 Subject: [PATCH 5/5] add barret reducer --- docs/algorithm/barret-reducer.md | 62 +++++++++++++++++++++++ docs/algorithm/img/barrett_flowchart.png | Bin 0 -> 226049 bytes docs/algorithm/img/barrett_reduction.png | Bin 0 -> 305903 bytes 3 files changed, 62 insertions(+) create mode 100644 docs/algorithm/barret-reducer.md create mode 100644 docs/algorithm/img/barrett_flowchart.png create mode 100644 docs/algorithm/img/barrett_reduction.png diff --git a/docs/algorithm/barret-reducer.md b/docs/algorithm/barret-reducer.md new file mode 100644 index 0000000..1d883a3 --- /dev/null +++ b/docs/algorithm/barret-reducer.md @@ -0,0 +1,62 @@ +# Barrett Reduction + +This module implements **Barrett reduction**, a mathematical technique used to speed up modulo operations on large integers. + +--- + +## 1. What is Barrett Reduction? + +In computer science, when working with very large integers (arbitrary-precision arithmetic), traditional division is one of the slowest operations. Calculating a simple remainder $x \pmod m$ normally requires a slow long-division step. + +Barrett reduction solves this by replacing the slow division with a combination of fast **multiplication** and **digit shifting** (limb-shifting). + +### The Core Concept: +To find the remainder of $x$ divided by $m$, we want to find the quotient $q = \lfloor x / m \rfloor$. Instead of dividing $x$ by $m$, we can multiply $x$ by the "inverse" of $m$ (i.e., $1/m$). + +Since we work with integers, we scale this inverse up to make it an integer. Let $b = 2^{64}$ represent our base (since each "limb" or array element in our code is a 64-bit integer), and let $k$ be the number of limbs in the modulus $m$. + +1. **Precomputation:** + We precalculate a scaling constant, $\mu$ (mu): + $$\mu = \lfloor \frac{b^{2k}}{m} \rfloor$$ + This is the only actual division we need to perform, and it is done once when initializing the reducer. + +2. **Estimating the Quotient:** + To calculate the modulo, we estimate the quotient $q$. Instead of full multiplication, we use a slightly adjusted formula to keep the numbers from growing too large: + $$q \approx \lfloor \frac{\lfloor x / b^{k-1} \rfloor \cdot \mu}{b^{k+1}} \rfloor$$ + In code, dividing by $b^{k-1}$ and $b^{k+1}$ is done using simple right-shifts of the array elements (limbs). + +3. **Calculating the Remainder and Correcting:** + Once we have our estimated quotient $q$, we calculate the remainder: + $$r = x - (q \cdot m)$$ + Because we used an approximation, our estimated quotient might be slightly smaller than the actual quotient (by at most 2). Consequently, the temporary remainder $r$ might still be larger than our modulus $m$. We correct this by subtracting $m$ from $r$ until $r < m$ (which is guaranteed to happen in at most two steps). + +--- + +## 2. Estimation and Final Correction + +To understand how the mathematical approximation interacts with the final adjustment loop, we can look at how the estimate behaves. + +### Visualizing the Process: +The following diagram illustrates how the algorithm behaves: + +![Quotient Estimation and Remainder Progress](img/barrett_reduction.png) + +* **Top Graph (Quotient Estimation):** Shows that the estimated quotient (due to rounding down during limb-shifts) occasionally falls slightly below the actual quotient. +* **Bottom Graph (Remainder Progress):** Shows the preliminary remainder $r$. Whenever the estimated quotient is slightly too small, the temporary remainder $r$ crosses the red line (the modulus $m$). In these instances, the correction loop subtracts $m$ until the remainder is brought within the correct bounds. + +--- + +## 3. Algorithm Flowchart + +The step-by-step control flow is represented in the diagram below: + +![Barrett Reduction Flowchart](img/barrett_flowchart.png) + +### Summary of Steps in Code: +1. **Shortcut check:** If the input $x$ is already smaller than the modulus $m$, return $x$ immediately. +2. **First shift:** Shift $x$ right by $k-1$ limbs to get $q_1$. +3. **Multiply:** Multiply $q_1$ by our precomputed value $\mu$ to get $q_2$. +4. **Second shift:** Shift $q_2$ right by $k+1$ limbs to get our final estimated quotient, $q_3$. +5. **Find remainder:** Calculate $r = x - (q_3 \cdot m)$. +6. **Adjust:** While $r \geq m$, subtract $m$ from $r$ (runs at most twice). +7. **Return:** Return the final reduced value $r$. \ No newline at end of file diff --git a/docs/algorithm/img/barrett_flowchart.png b/docs/algorithm/img/barrett_flowchart.png new file mode 100644 index 0000000000000000000000000000000000000000..e2b58c534a9051c4907338a46b61fcb17fd4e3d6 GIT binary patch literal 226049 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